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128 XIII INTEGRATION
(5) there exists an integrable function g± g: 0 such that, for all z e E, we
have \D2f(x, z)\ 5* 9i(x) almost everywhere in X.
Then h is differentiable at every point ofE, and we have
(126.96.36.199) h'(z) = f D2/(x, z) dtfx)
("differentiation under the.integral sign").
(iii) Now suppose that E is an open set in C and that f is a mapping of
X x E into C, satisfying the following conditions:
(6) for almost all x e X, the function zi>f(x, z) is analytic in E;
(7) for all z e E, the function x H-»/(JC, z) is integrable ;
(8) there exists an integrable function g^O such that for all z e E we
have \f(x9 z)| <j! g(x) almost everywhere in X.
Then h is analytic in E, and for each integer k > 0 and each point z e E,
the function x\~* D*/(^, z) fa integrable, and we have
(i) By (3.13.14) it is enough to show that, for every sequence (zn) tending
to z0 in E, we have lim h(zn) = A(z0); and this follows from the hypotheses
and from (13.8.4).
(ii) Again by virtue of (3.13.14), it is enough to show that, for each
sequence (O of real numbers =£0 and tending to 0, the sequence
has as limit the right-hand side of (188.8.131.52). But it follows from the hypotheses
and from (8.5.4) that
for all n} so that the result again follows from (13.8.4).
(iii) Let y : t\-+a + relt (0 <; t g 2n) be a circuit contained in E. For
each z in the interior of the disc |z a\ <£ r, we have
(184.108.40.206) f(x, z) = I /V*'M
V / «/ V^J **/ /^__- I f
~~" Zen h(z) = \f(x, z) dfi(x) is continuous at ZQ .